{"id":"d706fb8c-1ac2-4420-9bba-b70de51b9b64","arxiv_id":"2505.05121","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Empirical convergence rates for TDGF and DGM in Black-Scholes and Heston models show sampling stages and time steps are the main accuracy drivers, while sample count and network width have weak or unclear effects.","lead":"This paper benchmarks two neural network solvers for option pricing PDEs and measures how their error and training time respond to five training and architecture settings. It offers practical guidance on which settings improve accuracy and which add only cost.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reported convergence rates are not robust: no fitting procedure or uncertainty is given, and Table 1's Heston TDGF rate (-0.63) contradicts the 'almost linear' headline.","rationale":"The reader's weakest assumption is that the fitted rates in Tables 1-5 are meaningful scaling laws despite being computed on a fixed grid, averaged over five seeds, with no error bars and an unspecified fitting procedure. My stress-test confirms that this is the most load-bearing point: the headline findings are literally these fitted rates, and the paper offers no way to tell whether they are stable or artifacts of the fitting range and seed noise. I also identify a specific internal tension: the abstract and Section 1 say the L2-error decreases 'almost linearly' with sampling stages, but Table 1 reports -0.63 for TDGF in Heston, which is hard to describe as almost linear. Additionally, Section 5.5's plateau for the second-order scheme makes a single power-law fit over 2-25 time steps suspect. These issues do not invalidate the qualitative conclusions, and the paper is appropriately hedged in places, so no verdict change is needed; the conditional verdict stands. The concrete test I propose would settle whether the rates survive uncertainty quantification and range variation.","tokens_in":8780,"tokens_out":2983,"duration_ms":32787,"concrete_test":"Recompute the Table 1 sampling-stage exponents with (a) bootstrap resampling of the five seeds to obtain 95% confidence intervals and (b) leave-one-out fits on the parameter range (drop the smallest and largest N, and separately drop points beyond the stated plateau). If the TDGF-Heston exponent's confidence interval is wider than 0.3 or shifts by more than 0.2 under leave-one-out, the single power-law presentation and the 'almost linear' claim are not supported and should be replaced with range-bound, qualitative statements.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claims are the fitted exponents in Tables 1-5, but the paper does not state the fitting method (log-log least squares over which points? on seed-averaged errors?) and reports no standard errors or confidence intervals. The error measure is a deterministic L2-error on a fixed 47-point grid, averaged over five seeds, with no seed spread shown. This matters because several experiments mix regimes: Section 5.1 says error 'stops converging' beyond some sampling-stage count, and Section 5.5 says the second-order time-stepping 'does not improve any further' after 25 steps, yet Table 5 fits a single power law over the whole 2-25 range. A one-exponent fit across a decreasing-then-flat curve biases the rate toward zero and makes the fitted rate depend on the chosen range. Concretely, the abstract and Section 1 claim 'almost linear' decrease with sampling stages, but Table 1 gives TDGF-Heston -0.63, which is not close to the -1 that 'almost linear' implies. Since the recommendations (e.g., choosing sampling stages by speed/accuracy, choosing two or three layers) are derived from these rates, the missing uncertainty quantification and range sensitivity are load-bearing rather than cosmetic.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper empirically studies two deep PDE solvers, the Deep Galerkin Method (DGM) and the Time Deep Gradient Flow method (TDGF), for European call option pricing in the Black-Scholes and Heston models. Fixing one hyperparameter at a time, it measures the L2-error on a fixed 47-point grid against exact Black-Scholes prices and COS-based Heston reference prices, averages over five random seeds, and reports fitted 'convergence rates' (power-law exponents) for the number of sampling stages, number of samples, number of layers, number of nodes per layer, and, for TDGF, number of time steps and discretization order. It also reports training-time scaling for each parameter. The main findings are that the L2-error decreases roughly linearly with sampling stages, layers tend to reduce error without a clear rate, more time steps reduce error faster for the second-order scheme, and samples/nodes show no consistent effect.","tokens_in":9050,"tokens_out":7109,"duration_ms":67063,"significance":"If the reported rates were robust, the paper would give practitioners a useful accuracy-cost map for configuring DGM and TDGF in low-dimensional option pricing and would complement existing theory with concrete empirical guidance. The study has clear strengths: it uses external reference solutions (exact Black-Scholes and COS Heston), states its implementation choices in detail, and is appropriately hedged when no clear trend exists. Its main limitation is that the central quantitative claims, the fitted exponents in Tables 1-5, rest on an unspecified fitting procedure, no uncertainty quantification, only five seeds per configuration, and no released code. The practical recommendations in Section 6 are therefore only as strong as those fits allow.","major_comments":[{"comment":"The fitted convergence rates in Tables 1-5 are the central quantitative output of the paper, but the fitting procedure is never specified: the paper does not state whether a log-log least-squares fit was used, which data points were included, whether the fit was performed on the seed-averaged errors, or how the two-decimal precision of the exponents should be interpreted. No standard errors, confidence intervals, or seed-to-seed spreads accompany the rates, although the L2-error is reported only as an average over five seeds. In addition, the abstract and Section 1 claim that the error 'decreases almost linearly' with the number of sampling stages, yet Table 1 reports a TDGF-Heston rate of -0.63, which is far from the -1 implied by 'almost linear' and is not 'slightly larger than -1' as stated in Section 5.1. Because the recommendations in Section 6 are derived directly from these exponents, the missing specification and this internal contradiction are load-bearing.","section":"Section 5, Tables 1-5"},{"comment":"Section 5.5 states that after 25 time steps the second-order scheme 'does not improve any further', and Section 5.1 states that the error stops converging beyond some number of sampling stages, yet Table 5 and Table 1 report single power-law exponents fitted over the full ranges (K=2,...,25 and the full stage ranges). A single exponent fitted across a curve that first decreases and then flattens is biased toward zero and its value depends on the chosen range; the reported rates therefore cannot be interpreted as intrinsic convergence rates. The fits should be restricted to the decreasing regime, or piecewise rates should be reported, and the sensitivity of the exponents to the fitting range should be assessed.","section":"Section 5.5 and Section 5.1"},{"comment":"The 'convergence rates' for the number of layers and the number of nodes per layer are fitted from only four and five distinct parameter values, respectively (layers 1-4; nodes 10-50). Table 4 even reports positive rates for the DGM (0.15 for Black-Scholes, 0.07 for Heston), which the text does not reconcile with the claim that more nodes tend to decrease or not affect error. Without uncertainty quantification or a statement about fit stability over such short ranges, these exponents are over-interpreted; the only robust conclusion supported by the text is that there is 'no clear rate'. Please report the underlying data points and a stability check, or drop the rate interpretation for these parameters.","section":"Section 5.3 and Section 5.4"},{"comment":"The displayed DGM cost functional L(θ;t,x) is written as a sum of un-squared residuals, whereas the preceding text defines the DGM objective as the sum of the squared L2-errors of the PDE residual and the initial condition. As typeset, the formula minimizes a signed residual, which is not the DGM loss and could in principle be driven to negative values. If the implementation actually used the displayed objective, the DGM results correspond to a different training criterion; if the implementation used the squared form, the formula needs correction. Please clarify and fix.","section":"Section 2.2"}],"minor_comments":[{"comment":"The first sentence of Section 5.4 says 'We vary the number of layers from 10 to 50', which should read 'the number of nodes per layer'.","section":"Section 5.4"},{"comment":"In the bullet on sampling stages, 'with the the number of sampling stages' contains a duplicated article and should be corrected.","section":"Section 6"},{"comment":"In the bullet on nodes per layer, 'forty of fifty' should be 'forty or fifty'.","section":"Section 6"},{"comment":"The hardware description 'using one seventh instance of a NVidia Tesla A100 GPU' is unclear; please specify what 'one seventh instance' means (for example, a GPU slice or a specific resource allocation).","section":"Section 4"},{"comment":"In Section 5.2, the statement 'For the DGM the rates are larger' is ambiguous; 'less negative' would be clearer for readers comparing the fitted exponents.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The paper addresses a timely and practical question and is generally written clearly. The main concern is the reproducibility and robustness of the fitted rates; I recommend major revision, provided the authors can supply the fitting details, uncertainty quantification, and a reconciliation of the 'almost linear' claim with Table 1. The manuscript is within the scope of q-fin.CP, and the topic is appropriate for the journal. There is a noticeable reliance on the author's own prior work ([16], [18]) for the method and architecture; this is not inappropriate given the subject, but the novelty relative to those papers should be made explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Rou's paper is the first I've seen that maps L2 error and training time for DGM and TDGF across five hyperparameters in both Black-Scholes and Heston. That is genuinely useful engineering knowledge. The experimental setup is mostly sensible: exact Black-Scholes prices and COS prices for Heston as external references, a fixed 47-point grid, five seeds averaged, and the default parameter set clearly stated. The observations about training time scaling linearly with sampling stages, layers, and time steps are plausible and worth reporting. The citation pattern looks fine; the self-citations point to the actual architecture and the convergence theory used, which is legitimate.\n\nThe soft spots are load-bearing rather than cosmetic. The fitted convergence rates in Tables 1-5 are the central quantitative results, but the fitting procedure is never specified -- log-log least squares over which points? on seed-averaged errors? -- and no standard errors or seed spread are reported. The abstract and Section 1 claim the L2-error decreases 'almost linearly' with sampling stages, but Table 1 gives -0.63 for TDGF in Heston, which is not almost -1. Section 5.1 even says all rates are 'slightly larger than -1', which doesn't square with -0.63. Likewise, Section 5.5 says the second-order scheme stops improving after 25 time steps while Table 5 fits one power law over the whole 2-25 range; a decreasing-then-flat curve biases the exponent toward zero and makes the rate depend on the chosen range. Since the practical recommendations are derived from these rates, the missing uncertainty quantification is a real deficiency, not a stylistic one. No code or data is released, so the numbers can't be checked independently.\n\nThat said, this isn't a paper with a broken core. It is an honest, under-specified empirical study. The author is appropriately cautious in many spots, and the direction of the findings is probably right. The fix is straightforward: state the fitting method, add error bars or show seed trajectories, fit rates only on ranges where the error is actually decreasing, release code and data, and soften or qualify the 'almost linear' claim. With those changes I would trust the tables.\n\nWho is this for? Quants choosing hyperparameters for DGM or TDGF, and researchers benchmarking deep PDE solvers. I would not desk-reject it. It deserves a serious referee, but with major revision before publication.","headline":"Useful hyperparameter benchmark, but the headline convergence rates aren't yet credible: missing fitting details, no uncertainty, and 'almost linear' doesn't match Table 1.","tokens_in":9544,"tokens_out":2853,"would_cite":false,"duration_ms":27112,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91G20","91G60","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Deep PDE error shrinks almost linearly with sampling stages for option pricing","keywords":["option pricing","deep PDE solvers","Deep Galerkin Method","Time Deep Gradient Flow","empirical convergence","Black-Scholes model","Heston model","L2 error"],"falsifier":"Repeat the sampling-stages experiment with the same models and settings but evaluate the $L^2$ error on a much finer or adaptive grid and over a wider range, for example 16 to 2000 stages; if the log-log slope changes substantially or bends downward, the reported near-linear rate is an artifact of the evaluation grid or the chosen range rather than a robust scaling law.","tokens_in":1493,"feed_emoji":"📉","tokens_out":4045,"duration_ms":71045,"temperature":0.7,"pith_summary":"This paper measures how the accuracy of two deep-learning PDE solvers, the Deep Galerkin Method and the Time Deep Gradient Flow method, changes as their key training parameters are varied. It finds that the $L^2$ error falls almost linearly as the number of sampling stages grows, that adding layers helps but at an unclear rate, and that for the time-stepping method more time steps reduce error faster with second-order discretization than with first-order. The study also records training time, showing it grows linearly for stages, layers, and time steps but barely changes with the number of samples or nodes per layer. If these patterns hold, practitioners can choose training budgets based on a quantitative accuracy-versus-cost trade-off instead of guesswork.","feed_headline":"Deep PDE error shrinks almost linearly with sampling stages","feed_subtitle":"Measured accuracy-versus-cost curves for DGM and TDGF in standard option pricing now guide training budgets.","key_machinery":"The central objects are two neural-network PDE solvers applied to option-pricing problems. The Time Deep Gradient Flow (TDGF) method solves the pricing PDE time step by time step, minimizing an energy functional with Monte Carlo estimates of integrals and using either a first- or second-order time discretization. The Deep Galerkin Method (DGM) trains a single network over the whole time-space domain by minimizing the squared PDE residual plus a boundary term. The accuracy measure is the $L^2$ error on a fixed equidistant grid of 47 points per dimension, averaged over five random seeds, and convergence rates are obtained by power-law fits to the error as each parameter is varied one at a time.","core_discovery":"The paper's central finding is that empirical convergence of both solvers is driven mainly by sequential training effort: the $L^2$ error decreases almost linearly with the number of sampling stages, with fitted rates between roughly -0.6 and -0.9 across the Black-Scholes and Heston models, and training time rises linearly. Increasing the number of layers also reduces error, but the fitted rates vary and do not follow a clean law. For the Time Deep Gradient Flow method, increasing the number of time steps lowers the error, and the second-order scheme consistently converges faster than the first-order scheme in both test models. In contrast, the number of samples and the number of nodes per layer show no clear or uniform relationship with error, and for the Deep Galerkin Method the error can even rise slightly with more nodes per layer.","pith_inferences":["The near-linear rate in sampling stages resembles Monte Carlo averaging of the stochastic energy estimates; if so, variance-reduction techniques such as quasi-random sampling or importance sampling could push the empirical rate toward or past the observed -0.9.","The paper's fixed 47-point evaluation grid is coarse relative to the sharp payoff kink near the strike, so the reported rates may depend on the grid; evaluating on a finer or adaptive grid could reveal whether the scaling laws are genuine.","The contrasting behavior of samples and nodes between DGM and TDGF hints that DGM's error is limited by optimization rather than by approximation capacity, so better optimizers or learning-rate schedules might change the flat node-per-layer result."],"forward_implications":["For both solvers, increasing the number of sampling stages is the most dependable accuracy lever, and the near-linear error decrease means each additional stage buys roughly proportional accuracy until a plateau is reached.","The second-order time-stepping scheme for TDGF measurably outperforms first-order at the same time-step count, so using it is a direct accuracy improvement without changing the network architecture.","Since training time grows linearly with sampling stages, layers, and time steps, choosing these parameters is a clear speed-accuracy trade-off, and the reported rates give a basis for deciding where to stop.","The small training-time cost of additional samples and nodes per layer means, for TDGF, error can be reduced essentially for free by using many samples and many nodes.","The flat or slightly positive node-per-layer rate for DGM suggests that for this solver, network width beyond a modest size does not buy accuracy, so money is better spent on sampling stages."],"supporting_citations":[{"why":"Supplies the Black-Scholes pricing PDE and its exact solution, which serve as the benchmark for the experiments in the constant-volatility model.","marker":"[5]"},{"why":"Provides the COS reference prices for the Heston model, where no exact solution exists and which the neural-network solutions are compared against.","marker":"[8]"},{"why":"Introduces the time-stepping deep gradient flow method and the network architecture used in the TDGF experiments.","marker":"[18]"},{"why":"Introduces the Deep Galerkin Method that is the second solver under study and supplies the residual-minimization formulation the paper implements.","marker":"[19]"},{"why":"Provides the theoretical convergence analysis of TDGF that the empirical convergence rates are intended to complement or confirm.","marker":"[16]"},{"why":"Provides global convergence theory for DGM-style methods, giving a backdrop against which the observed DGM rates can be interpreted.","marker":"[14]"}],"fun_headline_variants":["Deep PDE error falls almost linearly with sampling stages","Sampling stages, not sample count, drive deep PDE solver accuracy","Option pricing deep solvers: error scales with training stages","TDGF second-order scheme converges faster in option pricing","Neural PDE solvers: linear error decay from sequential training"],"cache_read_input_tokens":11776,"weakest_assumption_plain":"The fitted convergence rates are meaningful scaling laws, which assumes that the $L^2$ error on a fixed grid of 47 points per dimension, averaged over five random seeds, is a stable and representative measure of solver accuracy, and that a single power-law fit over the chosen parameter ranges captures the true behavior.","fun_headline_variants_meta":{"raw":{"variants":["Deep PDE error falls almost linearly with sampling stages","Sampling stages, not sample count, drive deep PDE solver accuracy","Option pricing deep solvers: error scales with training stages","TDGF second-order scheme converges faster in option pricing","Neural PDE solvers: linear error decay from sequential training"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000196,"raw_usage":{"total_tokens":1319,"prompt_tokens":864,"completion_tokens":455,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":480,"completion_tokens_details":{"reasoning_tokens":374}},"tokens_in":480,"tokens_out":455,"duration_ms":4954,"temperature":1.0,"reasoning_tokens":374,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:11:30.750662+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the sampling-stages experiment with the same models and settings but evaluate the $L^2$ error on a much finer or adaptive grid and over a wider range, for example 16 to 2000 stages; if the log-log slope changes substantially or bends downward, the reported near-linear rate is an artifact of the evaluation grid or the chosen range rather than a robust scaling law.","supporting_citations":[{"cited_title":"Fang and C","cited_arxiv_id":null,"evidence_quote":"Provides the COS reference prices for the Heston model, where no exact solution exists and which the neural-network solutions are compared against."},{"cited_title":"Sirignano and K","cited_arxiv_id":null,"evidence_quote":"Introduces the Deep Galerkin Method that is the second solver under study and supplies the residual-minimization formulation the paper implements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the theoretical convergence analysis of TDGF that the empirical convergence rates are intended to complement or confirm."}],"review_version":1}